Graphics Reference
In-Depth Information
7.8 Inverse of a quaternion
As we have seen in earlier chapters, vectors can be multiplied together, but division does not
appear to have any meaning. Quaternions, on the other hand, can be divided, although the
division is represented as a product using an inverse quaternion. For example, q
−
1
is used to
represent 1
q, which means that
qq
−
1
=
1
(7.22)
In order to find q
−
1
, we multiply Eq. (7.22) by
q:
¯
qqq
−
1
¯
=¯
q
But from Eq. (7.21), we see that
2
¯
qq
=
q
Therefore,
2
q
−
1
q
=¯
q
and
q
¯
q
−
1
=
2
q
=
If
q
1, then
q
−
1
=¯
q
7.9 Rotating vectors using quaternions
Before discovering other features of quaternions, let's apply Eq. (7.16) to a practical example.
To recap, the equation to rotate a vector is
=
cos
2
sin
2x
i
z
k
0
v
cos
2
sin
2x
i
z
k
qv
q
¯
+
+
y
j
+
+
−
+
y
j
+
(7.23)
where
is the angle of rotation,
v
is the vector to be rotated, and
x
i
+
y
j
+
z
k
is the axis of rotation and in unit form.
The vector
v
is represented as a quaternion with a zero real component:
v
=
0
+
x
i
+
y
j
+
z
k
